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Counting is the process of determining the
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

number
of elements of a
finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
of objects, i.e., determining the
size Size in general is the magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term fo ...
of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a
unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...

unit
for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term ''
enumeration An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and r ...
'' refers to uniquely identifying the elements of a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
(combinatorial) set or infinite set by assigning a number to each element. Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...). There is archaeological evidence suggesting that humans have been counting for at least 50,000 years. Counting was primarily used by ancient cultures to keep track of social and economic data such as the number of group members, prey animals, property, or debts (that is,
accountancy Accounting or Accountancy is the measurement Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependent on the ...
). Notched bones were also found in the Border Caves in South Africa that may suggest that the concept of counting was known to humans as far back as 44,000 BCE. The development of counting led to the development of
mathematical notation Mathematical notation is a system of symbol A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective o ...
,
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
s, and
writing Writing is a medium of human communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area arou ...

writing
.


Forms of counting

Counting can occur in a variety of forms. Counting can be verbal; that is, speaking every number out loud (or mentally) to keep track of progress. This is often used to count objects that are present already, instead of counting a variety of things over time. Counting can also be in the form of
tally marks frame, Brahmi numerals (lower row) in India in the 1st century CE. Note the similarity of the first three numerals to the Chinese characters for one through three (一 二 三), plus the resemblance of both sets of numerals to horizontal tally ma ...

tally marks
, making a mark for each number and then counting all of the marks when done tallying. This is useful when counting objects over time, such as the number of times something occurs during the course of a day. Tallying is base 1 counting; normal counting is done in base 10. Computers use
base 2 Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM, an enterprise software company founded in 2009 with offices in Mountain View and Kraków, Poland *Base De ...
counting (0s and 1s), also known as
Boolean algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. Counting can also be in the form of
finger counting Finger-counting, also known as dactylonomy, is the act of counting using one's fingers. There are multiple different systems used across time and between cultures, though many of these have seen a decline in use because of the spread of Arabic num ...

finger counting
, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical operations. Finger-counting uses unary notation (one finger = one unit), and is thus limited to counting 10 (unless you start in with your toes). Older finger counting used the four fingers and the three bones in each finger (
phalanges The phalanges (singular: ''phalanx'' ) are digital Digital usually refers to something using digits, particularly binary digits. Technology and computing Hardware *Digital electronics Digital electronics is a field of electronics Elec ...

phalanges
) to count to the number twelve. Other hand-gesture systems are also in use, for example the Chinese system by which one can count to 10 using only gestures of one hand. By using
finger binary Finger binary is a system for counting Counting is the process of determining the number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object' ...
(base 2 counting), it is possible to keep a finger count up to . Various devices can also be used to facilitate counting, such as hand tally counters and
abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool that has been in use since ancient times and is still in use today. It was used in the ancient Near East The ancient Near East was the home of e ...

abacus
es.


Inclusive counting

Inclusive counting is usually encountered when dealing with time in
Roman calendar The Roman calendar was the calendar A calendar is a system of organizing days. This is done by giving names to periods of time, typically days, weeks, months and years. A calendar date, date is the designation of a single, specific ...
s and the
Romance language The Romance languages, less commonly Latin or Neo-Latin languages, are the modern languages that evolved from Vulgar Latin between the third and eighth centuries. They are a subgroup of the Italic languages in the Indo-European languages, Indo- ...
s. When counting "inclusively," the Sunday (the start day) will be ''day 1'' and therefore the following Sunday will be the ''eighth day''. For example, the French phrase for "
fortnight A fortnight is a unit of time equal to 14 day A day is approximately the period during which the Earth completes one rotation around its axis, which takes around 24 hours. A solar day is the length of time which elapses between the Sun reach ...
" is ''quinzaine'' (15  ays, and similar words are present in Greek (δεκαπενθήμερο, ''dekapenthímero''), Spanish (''quincena'') and Portuguese (''quinzena''). In contrast, the English word "fortnight" itself derives from "a fourteen-night", as the archaic "
sennight A week is a time unit equal to seven days. It is the standard time period used for cycles of rest days in most parts of the world, mostly alongside—although not strictly part of—the Gregorian calendar. In many languages, the days of the wee ...
" does from "a seven-night"; the English words are not examples of inclusive counting. In exclusive counting languages such as English, when counting eight days "from Sunday", Monday will be ''day 1'', Tuesday ''day 2'', and the following Monday will be the ''eighth day''. For many years it was a standard practice in English law for the phrase "from a date" to mean "beginning on the day after that date": this practice is now deprecated because of the high risk of misunderstanding. Names based on inclusive counting appear in other calendars as well: in the Roman calendar the ''nones'' (meaning "nine") is 8 days before the ''ides''; and in the Christian calendar
Quinquagesima Quinquagesima (), in the Western Christian Churches, is the last Sunday of Shrovetide, being the Sunday before Ash Wednesday. It is also called Quinquagesima Sunday, Quinquagesimae, Estomihi, Shrove Sunday, Pork Sunday, or the Sunday next before L ...
(meaning 50) is 49 days before Easter Sunday. Musical terminology also uses inclusive counting of between notes of the standard scale: going up one note is a second interval, going up two notes is a third interval, etc., and going up seven notes is an ''
octave In music Music is the of arranging s in time through the of melody, harmony, rhythm, and timbre. It is one of the aspects of all human societies. General include common elements such as (which governs and ), (and its associated co ...

octave
''.


Education and development

Learning to count is an important educational/developmental milestone in most cultures of the world. Learning to count is a child's very first step into mathematics, and constitutes the most fundamental idea of that discipline. However, some cultures in Amazonia and the Australian Outback do not count, and their languages do not have number words. Many children at just 2 years of age have some skill in reciting the count list (that is, saying "one, two, three, ..."). They can also answer questions of ordinality for small numbers, for example, "What comes after ''three''?". They can even be skilled at pointing to each object in a set and reciting the words one after another. This leads many parents and educators to the conclusion that the child knows how to use counting to determine the size of a set. Research suggests that it takes about a year after learning these skills for a child to understand what they mean and why the procedures are performed.Le Corre, M., Van de Walle, G., Brannon, E. M., Carey, S. (2006). Re-visiting the competence/performance debate in the acquisition of the counting principles. Cognitive Psychology, 52(2), 130–169. In the meantime, children learn how to name cardinalities that they can
subitize Image:Subitizing.svg, As the number of items increases, it becomes harder for an observer to instantly judge how many are present without counting. Subitizing is the rapid, accurate, and confident judgments of numbers performed for small numbers of ...
.


Counting in mathematics

In mathematics, the essence of counting a set and finding a result ''n'', is that it establishes a
one-to-one correspondence In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function (mathematics), function between the elements of two set (mathematics), sets, where each element of one set is paired with exactly on ...
(or bijection) of the set with the subset of positive integers . A fundamental fact, which can be proved by
mathematical induction Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement ''P''(''n'') holds for every natural number ''n'' = 0, 1, 2, 3, . . . ; that is, the overall statement is a ...
, is that no bijection can exist between and unless ; this fact (together with the fact that two bijections can be composed to give another bijection) ensures that counting the same set in different ways can never result in different numbers (unless an error is made). This is the fundamental mathematical theorem that gives counting its purpose; however you count a (finite) set, the answer is the same. In a broader context, the theorem is an example of a theorem in the mathematical field of (finite)
combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
—hence (finite) combinatorics is sometimes referred to as "the mathematics of counting." Many sets that arise in mathematics do not allow a bijection to be established with for ''any''
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
''n''; these are called
infinite set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
s, while those sets for which such a bijection does exist (for some ''n'') are called
finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s. Infinite sets cannot be counted in the usual sense; for one thing, the mathematical theorems which underlie this usual sense for finite sets are false for infinite sets. Furthermore, different definitions of the concepts in terms of which these theorems are stated, while equivalent for finite sets, are inequivalent in the context of infinite sets. The notion of counting may be extended to them in the sense of establishing (the existence of) a bijection with some well-understood set. For instance, if a set can be brought into bijection with the set of all natural numbers, then it is called "
countably infinite In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
." This kind of counting differs in a fundamental way from counting of finite sets, in that adding new elements to a set does not necessarily increase its size, because the possibility of a bijection with the original set is not excluded. For instance, the set of all
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s (including negative numbers) can be brought into bijection with the set of natural numbers, and even seemingly much larger sets like that of all finite sequences of rational numbers are still (only) countably infinite. Nevertheless, there are sets, such as the set of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, that can be shown to be "too large" to admit a bijection with the natural numbers, and these sets are called "
uncountable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
." Sets for which there exists a bijection between them are said to have the same
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and in the most general sense counting a set can be taken to mean determining its cardinality. Beyond the cardinalities given by each of the natural numbers, there is an infinite hierarchy of infinite cardinalities, although only very few such cardinalities occur in ordinary mathematics (that is, outside
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
that explicitly studies possible cardinalities). Counting, mostly of finite sets, has various applications in mathematics. One important principle is that if two sets ''X'' and ''Y'' have the same finite number of elements, and a function is known to be
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, then it is also
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and vice versa. A related fact is known as the
pigeonhole principle In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, which states that if two sets ''X'' and ''Y'' have finite numbers of elements ''n'' and ''m'' with , then any map is ''not'' injective (so there exist two distinct elements of ''X'' that ''f'' sends to the same element of ''Y''); this follows from the former principle, since if ''f'' were injective, then so would its restriction to a strict subset ''S'' of ''X'' with ''m'' elements, which restriction would then be surjective, contradicting the fact that for ''x'' in ''X'' outside ''S'', ''f''(''x'') cannot be in the image of the restriction. Similar counting arguments can prove the existence of certain objects without explicitly providing an example. In the case of infinite sets this can even apply in situations where it is impossible to give an example. The domain of enumerative combinatorics deals with computing the number of elements of finite sets, without actually counting them; the latter usually being impossible because infinite families of finite sets are considered at once, such as the set of
permutation In , a permutation of a is, loosely speaking, an arrangement of its members into a or , or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order o ...

permutation
s of for any natural number ''n''.


See also

* Automated pill counter *
Card reading (bridge) In contract bridge, card reading (or counting the hand) is the process of inferring which remaining cards are held by each opponent. The reading is based on information gained in the bidding and the play to previous tricks. The technique is used by ...
*
Calculation A calculation is a deliberate process that transforms one or more inputs into one or more results. The term is used in a variety of senses, from the very definite arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#A ...

Calculation
*
Cardinal number 150px, Aleph null, the smallest infinite cardinal In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ca ...
*
Combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...
*
Count data In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin wit ...
*
Counting (music) In music Music is the of arranging s in time through the of melody, harmony, rhythm, and timbre. It is one of the aspects of all human societies. General include common elements such as (which governs and ), (and its associated conce ...
*
Counting problem (complexity) In computational complexity theory Computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. ...
*
Developmental psychology Developmental psychology is the scientific Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern ...
*
Elementary arithmetic Elementary arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is sig ...
*
Finger counting Finger-counting, also known as dactylonomy, is the act of counting using one's fingers. There are multiple different systems used across time and between cultures, though many of these have seen a decline in use because of the spread of Arabic num ...

Finger counting
*
History of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...
*
Jeton Jetons or jettons are tokens or coin A coin is a small, flat, (usually, depending on the country or value) round piece of metal A metal (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( ...

Jeton
*
Level of measurement Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens Stanley Smith Stevens (November 4, 1906 – January 18, 1973) wa ...
* Mathematical quantity *
Ordinal number In set theory Set theory is the branch of that studies , which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of , is mostly concerned with those that ...
*
Particle number The particle number (or number of particles) of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All eve ...
* Subitizing and counting *
Tally mark frame, Brahmi numerals (lower row) in India in the 1st century CE. Note the similarity of the first three numerals to the Chinese characters for one through three (一 二 三), plus the resemblance of both sets of numerals to horizontal tally ma ...
*
Unary numeral system The unary numeral system is the simplest numeral system to represent natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this ...
*
List of numbers This is a list of articles about numbers. Due to the infinitude of many sets of numbers, this list will invariably be incomplete. Hence, only particularly notable numbers will be included. Numbers may be included in the list based on their mathe ...
* List of numbers in various languages * Yan tan tethera (Counting sheep in Britain)


References

{{Reflist Elementary mathematics Numeral systems Mathematical logic ca:Comptar